His name means "he who covers", and this probably refers to the sky. Varuna is the keeper of the cosmic order, a force called rta. It is rta which keeps everything working as it should, and Varuna's role as the one who governs rta makes him very important indeed. He is very closely linked to the god Mitra. Varuna is one of the Adityas and considered to be an asura, when those beings were still god-like and had not yet degenerated into demons. He is also associated with the moon and Soma, in Soma's incarnation as the drink of the gods. Varuna is seen as a white man in golden armor riding a Makara (a sea monster), holding a noose or lasso made from a snake.
Varuna is the keeper of the celestial waters, those which flow from the openings in the sky in the form of rain. He was worshiped with veneration and a healthy amount of fear, for as an asura Varuna did have his sinister aspects and was known to punish mortals who did not keep their word. He was the cosmic hangman and his usual method of punishment was to capture the offender with his noose. He was also a lord of the dead, a position he shared with Yama, and could confer immortality if he so chose.
In Vedic times, the worship of Varuna fell off as he was supplanted by Indra as king of the gods. One possible reason for this may go back to Indra's most famous exploit. When Vritra stole all the waters of the universe, the waters which Varuna was in charge of, it was Indra who had to fight the demon and get them back. It may have been because of this that Indra was able to supplant the overlordship of Varuna and become lord of the gods himself. Varuna then became god of the oceans and rivers; still important, but with hardly the grandeur he once had. The souls of those who drowned went to him, and he was attended by the nagas.